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Unit 6 Geometry Package MPM1D The basic elements of any geometric figure are points and lines. When lines intersect at a point, angles are formed. Q P Angles can be classified according to their measures in degrees. Triangles can be classified according to the measure of their angles. Triangles can also be classified according to the measures of their sides. Two angles whose measures add to 90o are called complementary. Two angles whose measures add to 180o are called supplementary. R Theorems in Euclidean Geometry The following is a list of “theorems” (rules) explored long ago by a Greek mathematician named Euclid. Theorem Opposite Angle Theorem (OAT) When two lines intersect the opposite angles formed are equal to each other. Angle Sum of a Triangle (AST) In any triangle, the sum of all 3 interior angles is equal to 180o Isosceles Triangle Theorem (ITT) Exterior Angle of a Triangle Theorem (EATT) In an isosceles triangle the two angles at the base of the equal sides are equal. Conversely, if two angles in a triangle are equal, the triangle is isosceles. In any triangle, the measure of an exterior angle is equal to the sum of the measures of the opposite two interior angles. Parallel Line Theorems A line that intersects two parallel lines is called a transversal. A transversal creates 8 angles. These angles have the following patterns. Alternate Angles Z-Pattern (PLT – Z) Corresponding Angles F-Pattern (PLT-F) Interior Angles C-Pattern (PLT-C) Problems 1. Find the value of x in each diagram. 2. Find measures in each of the following. (Be sure to give reasons for your answers) c) d) 80o x 3. Find the measures of angles <QPR and <QRP in each diagram below. 4. Find the missing measures. Give reasons. 38o 5. Find the measures of p and q in each figure below. 6. Find the missing measures in each figure. a) d) b) c) e) y z 7. Find the measures of all missing angles in the diagram below. Give reasons. Final Answers 1. a) 23o b) 109o c) 30o d)50o e) 22.5o f) 35o 2. a) x=110, y= 70 b)x= 50, y = 130 c) x = 62 d) x=115 3. a) 65o, 50o b) 70o, 70o 4. a) a = 65, b=50, c= 50 , d= 130, e = 130 b) a= 55, b = 55, c = 35, d = 110, e = 35 c) a = 42, b = 48, c = 41, d = 35 5. a) p = 60, q = 50 b) p = 40, q = 100 c) p = 54, q = 108 d) p = 21, q = 105 6. a) a = 40, b = 115, c = 105, d = 75, e = 65 b) a = 35, b = 145, c = 90, d = 90, e = 55 c) a = 56, b = 68, c = 56, d = 56 d) a = 55, b= 125, c=55, d = 125 e) y = 75, z = 75, x = 15 7. <AEB = 65o, <ACD = 65o, <ADC = 65o, <BAE = 50o, <AEF = 115o, <ADG = 115o, <CBE = 115o, <BED =115o, <FED = 65o,